to £264,000,000. They amount now to £2,120,000,000; and we have Peru, Austria, and other States, eight in number, with debts amounting to £3,200,000,000. I suppose that about the year 1822, the National Debt of England was about two thirds of the whole national debts of the world. If they amounted to £1,200,000,000, it would be about the outside, the debt of England being £800,000,000. There are now, I should think, between four and five thousand millions of public debts. Now, I apprehend that instead of England being, as it was, the principal debtor, owing about two thirds of the debts of the world, its debt is only about one sixth. France has now £900,000,000; at the battle of Waterloo it had only £64,000,000.

The CHAIRMAN-Mr. Baden's statement about approximations is very interesting, but it seems almost as if he got by intuition what was the result of his formula. I remember a remark of the late Mr. Peter Hardy's, who was a great advocate of approximations of all kinds. He once said to me that they are not only valuable, but more accurate than exact calculations, on account of the many conflicting elements which you have. In fact, even in calculating approximate annuities on three or four lives, you get better results by approximation than if you worked them out exactly. As to the result, it does not matter whether your calculations are a penny too much or a penny too little, but that they involve such enormous sums. I have been asked for a result for a bond of £100, and I have found afterwards that it was going to be for a loan of £100,000, or perhaps a million.

Mr. SUTTON, in reply, said-Recently I have had to make many calculations of this kind, and I find that there are cases where a ruleof-thumb method is not applicable.

Mr. BADEN-When I speak of the rule-of-thumb, I do not mean to say that I neglect any mathematical consideration on the subject. I merely mean that I use the direct method of trial. That is, I use the formula for an annuity, and try a series of rates.

Mr. SUTTON-Some loans are brought out in this way. Repayment does not commence until some years after the issue of the loan. That is a case where the ordinary method of approximation does not readily apply. It is all very well for gentlemen of great experience in these matters to use a tentative method like Mr. Baden's; but there are many people who only occasionally come across such questions, and it is for them, mostly, that formulas like these are wanted. I agree with Mr. Baden that in most questions of this kind the people asking for information do want the exact penny, the reason, as stated by our Chairman, being that the amounts dealt in are very large; and I am informed that even private individuals will sometimes take £100,000 or £150,000 worth of stock. Banks and stockbrokers frequently have inquiries made by their clients as to what rate of interest such and such an investment will yield. Indeed it would seem that private investors are getting as much interested in the matter as financiers. I have seen many prospectuses, but very few in which the actual rate of interest is put in. I believe that the common practise is that, in bringing out a loan, the financiers like to have these figures for their own guidance; but I have not seen them, as a rule, published in the prospectus itself. As to the remarks of Mr. Hodge

about the increase of these foreign debts, it certainly is very wonderful. Another point is also very striking, and I intend to bring it out more fully when the paper is completed. That is, how, notwithstanding the increase in the indebtedness of some countries, they are able from time to time to borrow, and to make fresh loans on easier terms than the preceding ones.

Negative Policy Values; being a paper read before the Actuarial Society of Edinburgh. By J. J. W. DEUCHAR, A.I.A. and A.F.A., of the Standard Life Assurance Company.

THE subject of Negative Policy Values has recently been touched

upon, more than once, in actuarial discussions, and its bearing upon the general question of insolvency is now beginning to be realized.

That the reserve made by many companies is reduced, by the appearance of some of their policy values as assets instead of as liabilities, is a fact well known, and difference of opinion no doubt exists as to the propriety of such a state of affairs. Although I believe it may be assumed, that the general opinion of competent judges is, that so long as a company is popular, and its valuations otherwise strong, the existence of a moderate amount of negative values among its assets is of small importance,-it is important to bear in mind that these negative values can only be realized under favorable circumstances, and at the first symptoms of unpopularity in a company they begin to disappear. Whenever, therefore, the popularity of an office is on the wane, the extent of negative policy values taken credit for by it, in its reserve, becomes a matter of vital importance. Or, in other words, to those companies which do not require assistance from such a source, credit may be given for negative values; but to those companies which do require such. assistance in showing sufficient funds, it is at least doubtful whether such credit should be given; and this, because the realization of negative values depends so entirely on the credit of the office.

I think I may say that opinions such as these are pretty generally held among experienced actuaries. It is, however, a matter of regret that no leading actuary (and, as far as I am aware, no one at all) has given the profession a thorough investigation into the nature and effects of negative policy values.

The intention of the present paper is, to consider, first, the nature of negative policy values, and under what circumstances these will arise; and second, to trace, if possible, the effect which the admission of these may ultimately have on the position of a company.

First, then, we have to consider the nature of negative policy

values; and, that we may do this more thoroughly, you will, perhaps, allow me to make a few remarks of an elementary nature, on three aspects of the value of a policy. These are, the value to the assured, the value to the office, and the value considered as the liability of the office.

The popular view of the value of a policy is a certain percentage of the premiums paid. Among the public generally, and even among the holders of life policies, there is not a very clear notion of what really constitutes the value of a policy. Indeed, I am not quite certain that all the members of the insurance profession itself have taken the trouble to inform themselves on this point. There prevails however, as I have said, the general idea that on surrendering a policy the assured should receive back a certain proportion of the premiums he has paid. Some offices guarantee that their surrender values will amount to a published percentage of the premiums paid, which fact has doubtless given rise to the popular idea on the subject; which is, as far as these offices are concerned, sufficiently correct. The actuary tells us that, in general, the assured pays for the risk run by an equal premium throughout life; whereas the risk-premium is an increasing quantity, less than the equal premium in the early years of assurance, and greater than it afterwards. The surplus of early years is retained and accumulated to make up for the deficiency of succeeding years; consequently, when a policy is surrendered, the necessity for accumulating this surplus ceases, and it may be handed over to the assured as the value of his policy. This is the theory of the popular view of the value of a policy. I need hardly say that under this view of the value of a policy, that is, considering it as a percentage of the premiums paid, no negative value can arise. From this we must infer that, to those companies which guarantee, from the outset, a certain percentage of the premiums paid, as surrender value, any system of valuation which produces negative results must be unsuitable. Also, although the company may guarantee a certain rate of value, the policy values, as brought out by the method of valuation used at an investigation, may possibly be less than this guaranteed value; consequently, care must be taken that a sufficient sum is reserved to meet demands for surrender value, the liability for which the company has assumed. Should this not be done, and many demands for surrender values be made, the company might find its Assurance Fund diminished, in so much greater ratio than its liabilities, as to render it insufficient to meet its claims. This, however, is only to

be feared where an office pursues the suicidal course of endeavouring to gain popularity by offering its policyholders greater inducements to surrender their policies than to continue them.*

The value of a policy to the office, is not a quantity which it is easy to assign definitely. In fact, the time of the actuary is so much taken up in examining the value of the policy to the assured, or the office's liability under it, that he is apt to forget this view of the subject altogether. Indeed it seems somewhat paradoxical to speak of the value to the office of its liabilities, but a little consideration will remind us that nearly all liabilities have a profitable side, and that, to incur liabilities with a view to profit, is the great principle involved in all mercantile transactions. The value of a policy to the office, then, consists in the probability of its becoming a source of profit, which is not assignable in an individual case, and is entirely prospective. That a company's policies will be sources of profit depends, 1st, on the mortality being more favorable than that anticipated, principally the result of the care and skill exercised in the selection of its lives, which leads to the company's receiving a greater number of premiums and more interest than are necessary to pay its claims; 2nd, on the rate of interest realized being higher than that assumed in its calculations, principally the result of care and skill exercised in the investment of funds; and 3rd, on the expenses being less than provided for,—the result of skill and economy in the management of the company's business.

It is rather curious to notice the resemblance between the operations in banking and insurance business. The bank receives a deposit of a certain sum, on condition of its paying a larger sum (namely, the deposit, and its interest at current rates) at a future time. The insurance company also receives a deposit of a certain sum (the single premium) on condition of its paying a larger sum (the sum assured) at a future time. In the case of the bank, the date of repayment is (or may be) fixed beforehand, while the sum to be repaid is left to be decided by the rates of interest which may have been current. In the insurance company, on the other hand, the sum to be repaid (the sum assured) is fixed beforehand, while the date of repayment is left to the prevailing rate of mortality to decide. A comparison of this kind is of course not to be overstrained, the similarity being more in form perhaps than in reality; but in view of the relation just noticed, I think we may

* This appears to be an important suggestion which deserves to be worked out more fully.-ED. J. I. A.

say, that the value of its policies to an insurance company is not dissimilar from that of its deposits to a bank.

The third aspect of the value of a policy, to which I wish to direct your attention, is that of the value considered as the liability of the office, and this view is the most important one in connection with our present investigation.

The contract of insurance is a twofold one, and consists of the obligation on the part of the company to pay the sum assured at death, provided the assured adhere to an assumed obligation on his part to pay the premium during life. The assured, however, is not under either legal or moral obligation to continue payment of the premium, and the effect of this on the transaction is, to give the assured the option of renewing his assurance at the end of each year, while it relieves the company of its liability the moment there is failure on the part of the assured to renew the contract by paying his premium. While, therefore, the full amount of the company's liability under a policy is that present sum which will accumulate to the sum assured at death, still, the sum assured can only be called for if the premiums have been regularly paid, the company need therefore only reserve the additional sum that is required over and above the future premiums, to provide payment of the sum assured. This we have expressed in the ordinary formula for the value of a policy, Ar+n-P(1+αx+n), Ax+n being the full amount of the company's liability, Px(1+ax+n) the value of the future premiums receivable, and Ax+n−Px(1+αx+n) the balance of the company's liability, or the sum it requires to lay past to supplement the future premiums, as they are received, in accumulating to the sum in the policy.

Now we know that, using net rates, Ar= P(1+ax), or that, at the commencement of the transaction, the present value of the sum assured is exactly equal to that of the future premiums. When, however, we consider the present value of the loaded premium, really payable by the assured, we have (Pa+K) (1+αx)>Ax; or Ax− (Px+K) (1+a) is a negative quantity. That is to say, (Px+K) (1+ax) the liability of the assured, in the matter of premiums, is greater than the liability of the office for payment of sum assured. This is what is meant by the negative value of a policy.

The general formula for the value of a policy, nVx=Ax+n -Px(1+αx+n), may be put into the very convenient form of Px+n(1+αx+n)−Px(1+αx+n), or (Px+n−Px)(1+αx+n). By the aid of the latter we are enabled to determine at a glance whether any particular data will produce negative results, if we have Pa,